CMS Rate Convexity Adjustment
A method is presented to calculating a particular multiplicative factor, which appears in a formula for a CMS rate convexity adjustment.
CMS Rate Convexity Adjustment
A method is presented to calculating a particular multiplicative factor, which appears in a formula for a CMS rate convexity adjustment. A CMS rate convexity adjustment provides a correction term to the forward CMS rate to match the mean value of the CMS rate under the forward probability measure.
We model the probability density of a CMS rate, under the forward swap measure, by a certain weighted sum of three log-normal densities. The defining parameter values for the log-normal densities above are determined by matching, in a least squares sense, the market price for various European style swaptions.
The model returns the ratio of the variance of the CMS rate, under the distributional assumptions above, to the variance of the CMS rate, instead assuming that it is log-normally distributed under the forward swap measure.
where
are unknown parameter values. The parameter values (1a-c) are determined by matching the model price for various European style swaptions, specified by respective
strike levels (in, at or out-of-the money),
diffusion,
and tenor,
Consider the fixed-for-floating rate swap defined above in Section 2.0. We seek to determine
where
Then
We now assume that
where
We calculate a ratio,
where
The convexity adjustment formula is then given by
We consider a fixed-for-floating interest rate swap specified by
forward start, 10 years,
maturity, 5 years,
The swap rate for the above is given by
References:
https://finpricing.com/lib/EqBarrier.html
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